Osweiler, 26, will open Thursday night's home game against the New Orleans Saints at FirstEnergy Stadium. Jackson said Cody Kessler will be second in line, followed by rookie DeShone Kizer and Kevin Hogan.
"At this stage in the evaluation process of our quarterbacks, we are going to have Brock start the preseason opener against the Saints on Thursday night," Jackson said in a statement released by the team. "A lot goes into our evaluation, but it's always going to be about efficiently and effectively running the offense. You want your starter to be able to do that despite any circumstance.
"Brock hasn't really gotten any first-team reps and this will give him that chance. We look forward to seeing what he can do with this opportunity throughout the week and against the Saints.''
Jackson emphasized any of the team's quarterbacks could earn the starting job for the season opener against the Pittsburgh Steelers.
"Cody will be the second quarterback to take the field, followed by DeShone and then Kevin,'' Jackson said Monday. "I'm pleased with the way our quarterbacks have competed and improved throughout the offseason and training camp.
"We are going to see a lot more of all of them this preseason and everyone in that room still has the chance to earn more opportunities and the right to lead our team as the starter when the regular season begins on September 10th."
The Houston Texans sent shockwaves through the NFL on the first day of free agency, sending a second-round pick in 2018 to Cleveland to get the Browns to take Osweiler off their hands and his $16 million guaranteed salary for 2017.
Houston signed Osweiler to a four-year contract worth $72 million with $37 million guaranteed last year, but he was benched and replaced by Tom Savage during the 2016 season only to get his job back when Savage was injured. For the season, Osweiler completed 59 percent of his passes, for 2,957 yards, 15 touchdowns, 16 interceptions and a passer rating of 72.2.
Osweiler told Cleveland.com last week that he is a better quarterback now than the one who went 5-2 for the Denver Broncos in 2015.
"I know that for a fact, because I've never been coached so hard on my fundamentals until I got here,'' Osweiler said. "Coach Jackson and (quarterback coach David) Lee, they do not let a single rep go without being on your fundamentals, and your fundamentals are what make you play great, so I am excited about the potential of the 2017 season.
"Like I said, not only do I feel capable of playing the way I did in 2015, I think I can play a lot better. Not a little better, a lot better. So I'm very excited about the potential this season.''